Representations of Coxeter groups and homology of Coxeter graphs

TL;DR

This paper classifies a new class of complex representations of Coxeter groups using graph homology, generalizing geometric representations and analyzing their structure through Kazhdan-Lusztig theory.

Contribution

It introduces a novel classification of Coxeter group representations via graph homology and describes the associated cell representations and their simple quotients.

Findings

Classified a new class of Coxeter group representations

Connected these representations to second-highest 2-sided cells

Identified simple quotients for specific Coxeter systems

Abstract

We classify a class of complex representations of an arbitrary Coxeter group via characters of the integral homology of certain graphs. Such representations can be viewed as a generalization of the geometric representation and correspond to the second-highest 2-sided cell in the sense of Kazhdan-Lusztig. We also give a description of the cell representation provided by this 2-sided cell, and find out all its simple quotients for simply laced Coxeter system with no more than one circuit in the Coxeter graph.